Exponential function series representation

We are immediately led to a faithful generalisation of function (13) if we retain the exponential function in s and expand the latter two factors in a series. The series contains the totality of powers of u, the even powers of t plus product terms. The exponential function takes care of the fact that the function vanishes at infinity. If such a representation is to be valid for each function, then powers of s have to be taken into the series expansion as well. As a final form of the eigenfunction we therefore put... 

The Fourier series representation assumes a known period 2P however, often we wish to analyze a function whose periodicity is unknown. From the exponential form of the Foiuier series, we obtain the Fourier transform by taking the limit 7 oo. We begin by writing the Fourier series as... 

The (HF) radial 3d function of a first transition series atom is compact and not accurately representable by a single exponential (r e 0- This is strikingly illustrated in Table III, which compares 3d orbital energies from single-zeta... 

The first paper in which the separability of the total infrared correlation function into its vibrational and rotational components was questioned is the paper by Van Woerkom, de Bleyser et al, (27). Their theory is based on use of the generalized cumulant expansion theorem for non-commuting quantum mechanical operators and involves the following assumptions, (i) The Hamiltonian of the liquid sample is written in the form H = E + F + G where E is the Hamiltonian for vibrational degrees of freedom, F is the bath Hamiltonian for rotational and vibrational degrees of freedom whereas G is the interaction Hamiltonian, (ii) G is small with respect to E + F. An interaction representation is used in which G is considered as a perturbation, (iii) The averaged ordered exponential is developped into a truncated cumulant expansion series. 

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